Richards proved that orientable, boundary-less, infinite type surfaces are completely
classified by their genus (possibly infinite), the space of ends \(E\) , and the subset of
ends accumulated by genus \(E^{G}\)  [1]}. When we talk about homeomorphisms between
subsets of \(E\) , we always assume they are type preserving. That is, we say \(U \subset E\) 
is homeomorphic to \(V \subset E\)  if there is a homeomorphism \(f \colon \,U \rightarrow V\)  such that \(f\) 
sends \(U \cap E^G\)  homeomorphically to \(V \cap E^G\) . Every homeomorphism of \(\Sigma \)  induces a
(type preserving) homeomorphism on the space of ends and, by Richards classification, every
homeomorphism of \(E\)  is induced by some element of \(\operatorname{Map} (\Sigma )\) .
